The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.
In celestial mechanics, the expression tidal force can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force^{ [2] } (for example, the perturbing force on the Moon): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.^{ [3] }
When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 4 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so-called tidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.^{ [4] } The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.^{ [5] } These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.
The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force.^{ [6] } The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of that astronomical body and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible.^{ [7] }
Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance, while the slope relative to value decreases in direct proportion to the distance. This is why the gradient or tidal force at any point is inversely proportional to the cube of the distance.
The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body.
For example, the Moon produces a greater tidal force on the Earth than the Sun, even though the Sun exerts a greater gravitational attraction on the Earth than the Moon, because the gradient is less.
Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference.^{ [7] }
As expected, the table below shows that the distance from the Moon to the Earth, is the same as the distance from the Earth to the Moon. The Earth is 81 times more massive than the Moon but has roughly 4 times its radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface.^{ [8] }
Gravitational body causing tidal force | Body subjected to tidal force | Diameter and distance | Tidal force | |||
---|---|---|---|---|---|---|
Body | Mass (m) | Body | Radius (r) | Distance (d) | ||
Sun | 1.99×10^{30} kg | Earth | 6.37×10^{6} m | 1.50×10^{11} m | 3.81×10^{−27} m^{−2} | 5.05×10^{−7} m⋅s^{−2} |
Moon | 7.34×10^{22} kg | Earth | 6.37×10^{6} m | 3.84×10^{8} m | 2.24×10^{−19} m^{−2} | 1.10×10^{−6} m⋅s^{−2} |
Earth | 5.97×10^{24} kg | Moon | 1.74×10^{6} m | 3.84×10^{8} m | 6.12×10^{−20} m^{−2} | 2.44×10^{−5} m⋅s^{−2} |
m is mass; r is radius; d is distance; 2r is the diameter G is the gravitational constant = 6.674×10^{−11} m^{3}⋅kg^{−1}⋅s^{−2}^{ [9] } |
In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon rotate about their common center of mass or barycenter, and their gravitational attraction provides the centripetal force necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.^{ [11] }
When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon.^{ [6] }
Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals,^{ [12] } and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.^{ [13] }
Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun. Tidal forces are also responsible for tidal locking, tidal acceleration, and tidal heating. Tides may also induce seismicity.
By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field.^{ [14] }
For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by vector subtraction of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the term tidal force is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.)
Tidal acceleration does not require rotation or orbiting bodies; for example, the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.
By Newton's law of universal gravitation and laws of motion, a body of mass m at distance R from the center of a sphere of mass M feels a force ,
equivalent to an acceleration ,
where is a unit vector pointing from the body M to the body m (here, acceleration from m towards M has negative sign).
Consider now the acceleration due to the sphere of mass M experienced by a particle in the vicinity of the body of mass m. With R as the distance from the center of M to the center of m, let ∆r be the (relatively small) distance of the particle from the center of the body of mass m. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass M. If the body of mass m is itself a sphere of radius ∆r, then the new particle considered may be located on its surface, at a distance (R ± ∆r) from the centre of the sphere of mass M, and ∆r may be taken as positive where the particle's distance from M is greater than R. Leaving aside whatever gravitational acceleration may be experienced by the particle towards m on account of m's own mass, we have the acceleration on the particle due to gravitational force towards M as:
Pulling out the R^{2} term from the denominator gives:
The Maclaurin series of is which gives a series expansion of:
The first term is the gravitational acceleration due to M at the center of the reference body , i.e., at the point where is zero. This term does not affect the observed acceleration of particles on the surface of m because with respect to M, m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆r is small compared to R, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration for the distances ∆r considered, along the axis joining the centers of m and M:
When calculated in this way for the case where ∆r is a distance along the axis joining the centers of m and M, is directed outwards from to the center of m (where ∆r is zero).
Tidal accelerations can also be calculated away from the axis connecting the bodies m and M, requiring a vector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆r is zero), and its magnitude is in linear approximation as in Figure 4.
The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about 1.1×10^{−7} g, while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about 0.52×10^{−7} g, where g is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon.^{ [16] } The solar tidal acceleration at the Earth's surface was first given by Newton in the Principia .^{ [17] }
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N). Force is represented by the symbol F.
Mass is the quantity of matter in a physical body. It is also a measure of the body's inertia, the resistance to acceleration when a net force is applied. An object's mass also determines the strength of its gravitational attraction to other bodies.
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces exceed the second body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit, material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenomena, and is measured in newtons per kilogram (N/kg). Equivalently, it is measured in meters per second squared (m/s^{2}).
Physical geodesy is the study of the physical properties of Earth's gravity and its potential field, with a view to their application in geodesy.
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The publication of the theory has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.
In classical mechanics, the gravitational potential at a location is equal to the work per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric potential with mass playing the role of charge. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.
In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body is the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.
A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation, ellipses and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by.
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.
In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes.
In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents
In physics, a force field is a vector field corresponding with a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field , where is the force that a particle would feel if it were at the point .
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum. This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies accelerate in vacuum at the same rate, regardless of the masses or compositions of the bodies; the measurement and analysis of these rates is known as gravimetry.
The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation and the centrifugal force . It is a vector (physics) quantity, whose direction coincides with a plumb bob and strength or magnitude is given by the norm .
tidal force.